11. Suppose that X(1), . . . , X(n) are the order statistics of a sample from a population having the rectangular distribution with p.d.f.

f(x) =

⎧⎪⎨

⎪⎩

0, x≤ 0 1

θ , 0 < x ≤ θ

0, x> θ

where θ is an unknown parameter. Show that for 0 < γ < 1, P



X(n) ≤ θ ≤

X(n)

n

√

1 − γ



= γ

(Note: n

√

1 − γ = (1 − γ)1/n).

288 7 Sampling Distributions